Theory of Smoluchowski

Peter Debye in 1944 further extended the work of Rayleigh and the fluctuation theory of Smoluchowski and Einstein to include the measurement of the scattering of light by macromolecular solutions for determining molecular size. 

Chemical reaction kinetics proceeds on the (often implicit) assumption that the reaction mixture is ideally mixed, and does not consider the time needed for reacting species to encounter each other by diffusion. The encounter rate follows from the theory of Smoluchowski. It turns out that most reactions in fairly dilute solutions follow chemical kinetics, but that reactions in low-moisture foods may be diffusion controlled. In the Bodenstein approximation, the Smoluchowski theory is combined with a limitation caused by an activation free energy. Unfortunately, the theory contains several uncertainties and unwarranted presumptions. 

To estimate x, the decrease in equilibrium adsorption and the actual adsorption rate according to the electrostatic phenomena, have to be considered. The application of Boltzmann s law assumes equilibrium condition of the DL and neglects any transport within the diffuse layer. Thus, the classic Boltzmann law cannot be used to describe the distribution of adsorbing ions within the double layer in non-equilibrium systems. The presence of any ionic flux is connected with a non-equilibrium state of the DL and the approach given by Overbeek (1943) in his theory of electrophoresis has to be considered. In that theory, the non-equilibrium of the DL causes non-linear dependencies of electrophoresis on the electrokinetic potential, in contrast to the theory of Smoluchowski where this effect is not allocated for. The importance of the non-equilibrium state of the DL for many other surface phenomena was emphasised by Dukhin Deijaguin (1974), Dukhin Shilov (1974), and Dukhin (1993). 

Consider the simplest situation of Figure 9.4a, where an absorbing spherical sink of radius R is at the center of the coordinate system and polymer chains are present in the solution around the sink. Following the classical theory of Smoluchowski (Chandrasekhar 1943), we assume that the sink absorbs the polymer chains as soon as the center-of-mass of a polymer chain approaches the surface of the sink. We identify the capture rate of polymer chains as the steady-state net flux of polymer chains into the absorbing sink. Let the initial number concentration of the polymer chains be cq. The polymer concenuation is continuously maintained as co at distances far from the sink. The polymer chains undergo only diffusion and there are no other convective contributions. The continuity equation for the number concentration of polymer chains in three... 

In HMC the momenta are constantly being refreshed with the consequence that the accompanying dynamics will generate a spatial diffusion process superposed on the ini rtial dynamics, as in BGK or Smoluchowski dynamics. It is well known from the theory of barrier crossing that this added spatial... 

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

Smoluchowski, M.V., 1917. Mathematical theory of the kinetics of coagulation of colloidal systems. Zeitschrift fur Physikalische Chemie, 92, 129-168. 

The collision frequency between drops may be estimated by means of Smoluchowski s theory (see, for example, Levich, 1962). The collision frequency (w = number of collisions per unit time per unit volume) for randomly distributed rigid equal-size spheres, occupying a volume fraction , is given by... 

The original theory of diffusional coagulation of spherical aerosol particles was developed by von Smoluchowski (1916,1917). The underlying hypothesis in this theory is that every aerosol particle acts as a sink for the diffusing species. The concentration of the diffusing species at the surface of the aerosol particle is assumed to be zero. At some distance away, the concentration is the bulk concentration. 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

The central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

This expression is a fundamental result of the theory of bulk ion recombination and has been extensively used in interpreting experimental results of diffusion controlled reactions. The Debye-Smoluchowski expression can also be written in terms of the mobility,... 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

A more quantitative measure of stability, known as the stability ratio, can be obtained by setting up and solving the equation for diffusive collisions between the particles. Quantitative formulations of stability, known as the Smoluchowski and Fuchs theories of colloid stability, are the centerpieces of classical colloid science. These and related issues are covered in Section 13.4. 

As mentioned above, in 1917 M. Smoluchowski applied the theory of diffusion to this situation to evaluate the rate of doublet formation. According to Fick s first law (Equation (2.22)) J, the number of particles crossing a unit area toward the reference particle per unit of time is given by... 

What is meant by rapid coagulation What is the basic principle behind the Smoluchowski theory of rapid coagulation What is the rate coefficient for rapid coagulation How is it defined, and what properties of the dispersion determine its magnitude What are the limitations of this theory as presented in the text ... 

Values of the work functions of different faces are best known for tungsten. A review of thermionic measurements and their theoretical interpretation is given by Herring and Nichols (7a). The reader interested in the theory of the phenomenon is also referred to Smoluchowski (7b). Work functions of two crystallographic directions in tungsten have recently been determined by field emission technique by Drechsler and Miiller (7c). 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

The weakness of this boundary condition is being able to justify a large enough distance to be comparable with an infinite distance, or perhaps 1000R. In practice, this would require B to be in excess over A by about 109 times A more reasonable approach to this outer boundary condition would be to require that there be no loss or gain of matter over this boundary, as there is an approximately equal tendency for the B reactants to migrate towards either A reactant upon each side of the boundary. The proper incorporation of this type of boundary condition into the Smoluchowski model leads to the mean field theory of Felderhof and Deutch [25] and is discussed further in Chap. 8 Sect. 2.3 and Chap. 9 Sect 5. 

The Smoluchowski theory of diffusion-limited (or controlled) reactions relies heavily on the appropriateness of the inital condition [eqn. (3)]. Though the initial condition does not determine the steady-state rate coefficient [eqn. (20)] because diffusion of B in towards the reactant A is from large separation distances (>10nm) in the steady-state, it does directly determine the magnitude of the transient component of the rate coefficient because this is due to an excess concentration of B present initially compared with that present in the steady-state. As a first approximation to the initial distribution, the random distribution is intuitively pleasing and there is little experimental evidence available to cast doubt upon its appropriateness. Section 6.6 and Chap. 8 Sect. 2.2 present further comments on this point. 

The remainder of this section considers several experimental studies of reactions to which the Smoluchowski theory of diffusion-controlled chemical reaction rates may be applied. These are fluorescence quenching of aromatic molecules by the heavy atom effect or electron transfer, reactions of the solvated electron with oxidants (where no longe-range transfer is implicated), the recombination of photolytically generated radicals and the reaction of carbon monoxide with microperoxidase. 

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